The present invention is directed to a web or support structure, and more particularly to a web or support structure that could be utilized to form structural elements.
Architects, civil and structural engineers conventionally utilize various web structures for supporting, for example, trusses, floors, columns, etc. Typically, web structures form various lattices or framework that support underlying or overlying supports. In this regard, structural engineers are quite familiar with a “Fink truss” (FIG. 2), the geometry of which encodes an approximation of a “Sierpinski triangle”—the “limit” of the recursive design indicated in FIGS. 1-3.
It has been observed in the past that the geometry of the hardest substance known to man, namely diamonds, and the modern roof truss encode and represent the approximations to certain fractals. The Fink truss (FIG. 2) is an engineering design that is a level-1 2-web. In nature, carbon-carbon bonding in diamond encodes a level-1 3-web.
In my earlier U.S. Pat. No. 6,931,812, which is hereby incorporated herein in its entirety by reference, I disclosed a 4-web structure represented in a 3-dimensional space that at a level-0 contains ten triangles.
The 2-web and 3-web date to circa 1900, while the 4-web from the 4th dimension was realized within human vision late in the 19th Century, and eventually published in the literature circa early 2003 (Reference No. 3). The 4-web is pictured on the cover of my book (Reference No. 2). Each of the 2-web, 3-web, 4-web, and 5-web are concrete examples of an abstract space that is referenced in the literature as “Lipscomb's Space” that I invented to solve a half-century old problem in dimension theory.
Page 20 of my book (Reference No. 2) contains mathematical details about the lower dimensional webs. In particular, the “ω with superscript 5” notation in the book denotes the 5-web, and the “J with subscript 6” notation also denotes the 5-web, where the 6=5+1 indicates the number of vertices of the 5-web. In general the “ω with superscript n” denotes an n-web and the “J with subscript n+1” also denotes the n-web, where n+1 indicates the number of vertices of the n-web.
Simply put, it has been an open problem to create a picture of an approximation to a 5-web within 3-space (human visual space). In the present disclosure, I use hyperbolic geometry to show how to visualize within 3-space (human visual space) such approximations to the 5-web. Topologically speaking, the new concept extends the 4-web design. Perhaps more importantly, however, is the fact that curved hyperbolic segments in the 5-web may induce more microscopic movement than the straight segments now used, for example, in the 4-web medical implants. Such result is likely to increase bone growth because the growth rate of bone is evidently increased by microscopic flexing of the 4-web segments.
Recalling again the value of “triangles” when it comes to designing high-strength structures, let us also recall that the 3-web level-0 (FIG. 4) has six struts and four triangles. The strength increases as the number of triangles increases. For example, I have shown in my unpublished article (Reference No. 1), that the addition of a single polar strut (compare FIG. 4 to the top half of FIG. 7), could increase compressive strength by as much as 20%. That is, the polar strut provides more triangles.
In order to understand the new 5-web design (subject of this application), recall that the “4” in “4-web” refers to the “4th-dimension”—the place where the 4-web originally existed. There are also “2-webs”, which exist in 2-dimensional planes, and “3-webs”, which exist in 3-dimensional space (human visual space). Mathematically, this list of webs and corresponding dimensions continues ad infinitum. Sample illustrations of the webs existing in lower-dimensional space are shown in FIGS. 1-9.
More specifically, FIGS. 1-3 depict “levels” of 2-webs. Specifically, FIGS. 1-3 show a “level-0” (a single triangle), a “level-1” (three level-0 2-webs, illustrated as red, green, and blue), and a “level-2” 2-web (containing three level-1 2-webs), respectively. As the level-numbers increase, these structures approach a “limit”, which is called the “2-web”.
FIGS. 4-6 depict “levels” of 3-webs. Specifically, FIGS. 4-6 show a “level-0” (a single tetrahedron), a “level-1” (four level-0 3-webs, illustrated as red, green, blue, and gold), and a “level-2” 3-web (containing three level-1 3-webs), respectively. As the level-numbers increase, these structures approach a “limit”, which is called the “3-web”.
FIGS. 7-9 depict “levels” of 4-webs. Specifically, FIGS. 7-9 show a “level-0” (a single hexahedron), a “level-1” (five level-0 4-webs, illustrated as red, green, blue, gold, and black), and a “level-2” 4-web (containing five level-1 4-webs), respectively. Again, as the “level numbers” increase these structures approach a “limit” that is called a “4-web”.
The key is to observe the inductive process, illustrated in FIGS. 1-9. The “inductive process” is a process that allows us to start at a given level, and then move to the next level using the given level. In more detail, the process is a two-step process. First, congruent copies of a given level are made. Second, these congruent copies are positioned so that each is just touching the others. To say that two congruent structures are “just touching” is to say that there exists one and only one point that is contained in both structures.
For example, consider the inductive process illustrated in FIGS. 4-6. We start with a tetrahedron (FIG. 4- four vertices), which is a level-0 3-web. Then, we create four congruent copies (colored red, green, blue, and gold). Next, we position these four copies so that each is just touching the other three. This positioning is shown in FIG. 5. To find the just-touching points, simply seek the points where two distinct colors meet. In particular, find the point where the red congruent copy meets the green congruent copy. That point is the “just touching point” for those copies. The construction of congruent copies followed by the “just touching” positioning allows one to move from one level to the next to infinity. Such an algorithm defines the inductive process.
In summary, the Fink truss (FIG. 2), which is a level-1 Sierpinski triangle, has been utilized for many years in constructing various support structures. To date, diamond which has the geometry of a level-1 Sierpinski cheese as its basic building structure is known to be the hardest structure. In the present invention, I have discovered a geometrical structure that represents the next step over the 4-web structure, i.e., the 5-web structure.